Lattice points, perturbation theory and the periodic polyharmonic operator.
The authors consider the polyharmonic operator acting in \(L_2(\mathbb R^d)\), \(d>1\), perturbed by a real-valued periodic function: \(H=(-\Delta )^l+V\), \(l>0\). It is known that the spectrum of \(H\) is formed by closed intervals (spectral bands), possibly separated by gaps. The paper is devoted to the aspect of this structure known as the Bethe-Sommerfeld conjecture, which states that the number of spectral gaps is finite when the dimension \(d\) is higher than one. The conjecture has been a subject of intense study for a number of authors. In particular, the first proof for a physically interesting case \(l=1\), \(d=3\) was given in [\textit{M. M. Skriganov}, Invent. Math. 80, 107--121 (1985; Zbl 0578.47003)]. The proof of the conjecture is getting more challenging when dimension \(d\) grows, \(l\) staying fixed, since the structure of the spectrum is getting more complicated for larger dimensions. The strongest result achieved so far is the result of the authors from a previous paper [\textit{L. Parnovski} and \textit{A. V. Sobolev}, Duke Math. J. 107, 209--238 (2001; Zbl 1092.35025)]. They proved the conjecture in the case \(6l>d+2, d>1\). The goal of the present paper is to loosen this condition further. In the present paper the conjecture is proven for the case \(8l>d+3, d>1\). The approach used by the authors comprises two main ingredients. The first is number-theoretic estimates, more precisely, estimates on the number of lattice points inside a ball of a large radius. The second is an estimate on the difference between the counting functions of the perturbed and unperturbed problems. All the necessary number-theoretic facts were obtained in the authors' previous paper [op. cit.] and are used in the present paper without modifications. On the contrary, for the second ingredient the authors now rely on a new bound obtained in [\textit{Yu. E. Karpeshina}, Ark. Mat. 38, no. 1, 111--137 (2000; Zbl 1021.35027)]. The last modification enables the authors to improve the sufficient condition for the validity of the conjecture from \(6l>d+2\) to \(8l>d+3\).
scientific article; zbMATH DE number 1684725
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